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Chapter 9: Problem 5

Find the differential \(d y\). \(y=\sqrt{9-x^{2}}\)

Short Answer

Expert verified
The differential \(dy\) is \(-\frac{x dx}{\sqrt{9-x^{2}}}\).

Step by step solution

01

Rewrite the function

The given function is a composite function and it might need a little rewrite before differentiation is performed. Let's rewrite the function as, \(y=(9-x^{2})^{0.5}\)
02

Apply the Chain Rule

Applying the chain rule, the outer function is \(u^{0.5}\) and the inner function is \(u = 9 - x^{2}\). First differentiate the outer function with respect to \(u\) and then differentiate the inner function with respect to \(x\).\n Derivative of the outer function \(u^{0.5}\) with respect to \(u\) is \(0.5u^{-0.5}\) and derivative of the inner function \(9 - x^{2}\) with respect to \(x\) is \(-2x\).
03

Write down the differential

Substituting these into the chain rule gives \(\frac{d y}{d x} = 0.5*(9-x^{2})^{-0.5} * (-2x)\). This can be further simplified to \(\frac{d y}{d x} = -\frac{x}{\sqrt{9-x^{2}}}\). Representing dy gives dy = \(\frac{d y}{d x} dx = -\frac{x dx}{\sqrt{9-x^{2}}}\).

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Most popular questions from this chapter

The concentration \(C\) (in milligrams per milliliter) of a drug in a patient's bloodstream \(t\) hours after injection into muscle tissue is modeled by $$ C=\frac{3 t}{27+t^{3}} $$ Use differentials to approximate the change in the concentration when \(t\) changes from \(t=1\) to \(t=1.5\).

The cost function for a certain model of personal digital assistant (PDA) is given by \(C=13.50 x+45,750\), where \(C\) is measured in dollars and \(x\) is the number of PDAs produced. (a) Find the average cost function \(\bar{C}\). (b) Find \(\bar{C}\) when \(x=100\) and \(x=1000\). (c) Determine the limit of the average cost function as \(x\) approaches infinity. Interpret the limit in the context of the problem.

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4}-4 x^{3}+16 x\)

The profit \(P\) for a company producing \(x\) units is \(P=\left(500 x-x^{2}\right)-\left(\frac{1}{2} x^{2}-77 x+3000\right)\) Approximate the change and percent change in profit as production changes from \(x=115\) to \(x=120\) units.

A state game commission introduces 50 deer into newly acquired state game lands. The population \(N\) of the herd can be modeled by \(N=\frac{10(5+3 t)}{1+0.04 t}\) where \(t\) is the time in years. Use differentials to approximate the change in the herd size from \(t=5\) to \(t=6\).
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